鲁洛克斯三角形上的Schwarz边值问题

Schwarz Boundary Value Problem on Reuleaux Triangle

摘要: 本文给出了鲁洛克斯三角形上一类特殊函数——其实部在边界上是循环对称函数——的Schwarz边值问题的解法，首先根据这类函数的特殊性将原问题进行改造，使其分别转化为实部在边界上关于实轴对称的Schwarz边值问题和实部在边界上关于实轴反对称的Schwarz边值问题，我们发现改造后的两个问题解之和就是原问题的解，然后通过对称扩张将改造后的两个问题转换为两组Riemann边值问题，最后通过求解这两组Riemann边值问题得到原问题的解。

Abstract: This paper presents the approach to the Schwarz boundary value problem of a special function on the Reuleaux triangle, the real part of this function is a cyclosymmetric function on the boundary. According to the particularity of this type of function, the original problem is first transformed into a Schwarz boundary value problem with the real part symmetric about the real axis on the boundary and the real part antisymmetric Schwarz boundary value problem with the real axis on the boundary. The sum of the solutions of these two problems is equal to the solution of the original problem. After that, the two problems after the conversion are further converted into two sets of Riemann boundary value problems through the symmetric expansion, finally, the solution of the original problem is obtained by solving the solutions of the two sets of Riemann boundary value problems.